This is a 22-page exploration of the inverse relationship between multiplication and division as it relates to both multiplication and division with fractions.
It begins with a discussion of the concept of inverse. Students gain familiarity with the terms Invert, inverse, and reciprocal by writing “inverse” and “reciprocal” expressions and numbers.
Beginning with unit fractions to clarify the concepts, students see the connection between division by a whole number and multiplication by a fraction, and make sense out of the “Invert the divisor and multiply” procedure. They also see how the inverse relationship is a part of everyday math both in and out of the classroom, and how they have already been employing the procedure when multiplying whole numbers by fractions.
They also correlate verbal expressions of division with the mathematical notation.
For instance, the expression “Four divided by one-third” is asking, “How many thirds fit into 4?” This relationship is easily illustrated with Fraction Towers, but there are 12 pages of images of fraction bars modeling equivalent fractions and fractional division in the same way as Fraction Towers. These pages would be useful as review for students even if they use Fraction Towers in doing the activity.
Students practice dividing whole numbers by proper fractions, fractions by whole numbers, and fractions by fractions, including problems in which the quotient is a mixed number.
Six pages of word problems are included which can be used to introduce different sections of these pages. There are word problems requiring division of whole numbers with fractional divisors, fractions with whole number divisors, and fractions with fractional divisors.
Common Core Standards:
6.NS.1: “Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions...”
5.NF.7: “Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions...”
5.NF.7.a. “Interpret division of unit fraction by a non-zero whole number, and compute such quotients...”
5.NF.7.b. “Interpret division of a whole number by a unit fraction, and compute such quotients...”
5.NF.7.c. “Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions...”
Subject: Math, Fractions, Word Problems
Level: Grades 5 – 6
Length/Duration: 22 pages of student work plus 12 pages of fractional number lines, 3-5 days
A Smart Notebook 14 file is also included.